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Questions and Answers
What is phat?
Sample proportion
What is the sample mean represented by?
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What does the sample proportion represent?
p hat
What is the population proportion represented by?
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What would the population distribution be if 30% of the population has at least one cat?
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What would the sample/data distribution be for a sample of 100 if 27 said they have at least one cat?
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What does phat estimate in relation to the population proportion?
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If each student in the class took their own sample and reported the results, what would occur?
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What is the standard error of the distribution of phat?
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What are the conditions for the sampling distribution of phat to be approximately normal?
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What is the mean of the sampling distribution of phat?
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In the scenario where 54% of registered voters support a candidate, can we use the normal distribution to approximate probabilities?
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What is the probability that in a sample of 400, at least half support a new Walmart super center?
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Is the average salary of a random sample of 5 lawyers from a firm less than $100,000 likely?
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What is necessary to determine the sample size needed for a study?
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The margin of error for CI for p is _____
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What factors influence the size of the sample needed for a particular margin of error?
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What is the sample size needed if you have no clue about the proportion of students financially supported by their parents?
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What is the sample size needed if a previous study found the proportion was near 0.9?
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What is the formula for calculating the margin of error for confidence intervals for the population mean?
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What is the formula for determining the minimum sample size needed for achieving a desired margin of error?
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How many 1 oz containers of water should be collected for estimating the average pH with a margin of error of 0.03 at 95% confidence?
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How many students should be sampled to estimate the true average GPA within +/- 0.05 points with 99% confidence?
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Can you use the small sample method for confidence intervals if you don't have 15 successes and 15 failures?
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What is the primary parameter being estimated in the chemotherapy patient study?
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What is the confidence interval for the parameter in the chemotherapy patient study?
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Why is the width of the confidence interval for reduced queasiness so wide?
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What can be done to achieve a smaller interval while maintaining 95% confidence?
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What is the bootstrap confidence interval?
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What percentiles are used to compute the 90% confidence interval in the bootstrap method?
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What was the average weight of Murphy in pounds from the provided data?
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What are the two types of statistical inference?
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What is the sampling distribution of the sample proportion phat given p=0.5 and n=200?
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What is the probability of observing a result as high or higher than 0.57 given p=0.5?
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Study Notes
Sampling Proportions and Means
- P-Hat (p̂) represents the sample proportion, calculated from binary data.
- Sample Mean (x̄) denotes the average value of quantitative data, key for analysis.
- Population Proportion (p) is the true proportion in the entire population.
Example of Proportions
- If 30% of a population has at least one cat (p = 0.30) and a sample of 100 yields 27 with a cat,
- Population Distribution: X = 0 (no cats), 1 (at least one cat); Probability Histogram: p(1) = 0.3, p(0) = 0.7.
- Sample Distribution: p̂ = 27/100 = 0.27, variability expected between samples.
Sampling Distributions
- Sampling Distribution: Probability distribution outlining all possible values a statistic can assume.
- Normal Approximation: The distribution of p̂ approximates normality under conditions np > 15 and n(1-p) > 15.
- Standard Error of p̂: Calculated as √[p(1-p)/n].
Central Limit Theorem (CLT)
- As sample size increases (n ≥ 30 generally suffices), the sampling distribution of x̄ approaches normality regardless of the original population distribution.
Conditions for Normality
- For sample proportion (p̂): Focus on fulfilling the criteria of np > 15 and n(1-p) > 15.
- For sample mean (x̄): Needs either a normal distribution or n ≥ 30.
Confidence Intervals
- Constructing confidence intervals involves using the sample mean (x̄) and standard error to estimate the population parameter.
- For instance, with 1262 respondents, a 95% confidence interval for daily relaxation hours was (3.42, 3.70).
Sample Size Determination
- Sample size can be calculated based on desired margin of error (m), confidence level (z), and estimated proportion (p̂).
- A higher estimated proportion near 0.5 typically requires a larger sample size for precision.
Special Situations
- For lawyer salaries with a sample mean under uncertainty, given only a small sample size (n=5), sample mean conditions may not hold if the population distribution isn’t normal.
- Calculating probabilities related to normal distributions often involves finding z-scores and using normal probability tables.
Practical Examples
- For example, assessing at least half a sample supporting a proposal relies on computing probabilities through z-scores derived from sample proportions.
- Estimating the average GPA with the intent of +/- 0.05 accuracy requires computing a necessary sample size based on standard deviation.
General Guidelines
- Always round up to the nearest whole number when determining sample sizes.
- Employing prior studies' estimates for p can significantly affect necessary sample sizes, especially when unknown.
Key Definitions
- Margin of Error for Proportion: m = z√[p̂(1-p̂)/n]
- Margin of Error for Mean: m = t(s/√n)
- The mean of the sampling distribution of x̄ will equal the population mean (μ).
- Sample means (x̄) tend to form a normal distribution as sample size increases, centered around the population mean, becoming less spread.
Important Notes
- Normal distributions allow for easier calculations of probabilities and making inferences about sample and population parameters.
- Ensure conditions for the sampling distributions are satisfied prior to using normal approximation methods.### Sample Size Calculation
- Required sample size ( n ) for 99% confidence level is calculated as ( n=(.67)^2(2.58)^2/.05^2 \approx 1196 ).
- Estimated standard deviation is derived from the range of data divided by 6; for a range of 4, it equals ( \frac{4}{6} = \frac{2}{3} ).
Small Sample Method for Confidence Intervals
- When fewer than 15 successes and failures are present, apply a correction by adding 2 successes and 2 failures in calculations.
- Adjust ( \hat{p} ) using the formula ( \hat{p} = \frac{x+2}{n+4} ).
- Utilize the same confidence interval formula as for large samples, substituting ( n ) with ( n+4 ) in the standard error calculation.
Case Study on Chemotherapy Patients
- Study involved 16 parents who reported reduced queasiness after using a new ginger-based drug; 10 patients reported effectiveness.
- The parameter estimated is the population proportion rather than the population mean.
- Assumptions for a confidence interval are not met due to insufficient successes and failures.
- The true proportion ( p ) of patients with reduced queasiness is estimated using the confidence interval formula.
Confidence Interval Construction
- Calculated ( \hat{p} = \frac{12}{20} = 0.6 ).
- The 95% confidence interval is ( 0.6 \pm 1.96 \sqrt{\frac{0.6(1-0.6)}{20}} ) yielding the interval ( (0.39, 0.81) ).
- Interpretation: There is a 95% confidence that this interval captures the true proportion of chemotherapy patients experiencing reduced queasiness.
- The wide interval reflects variability due to the small sample size.
- To reduce interval width while maintaining 95% confidence, increase the sample size.
Bootstrap Confidence Interval
- Bootstrap methods allow determination of confidence intervals when traditional formulas are unavailable, particularly for median or standard deviation.
Computing Bootstrap Confidence Intervals
- Resampling is performed with replacement from the original dataset, generating numerous sample statistics.
- A 95% confidence interval is determined by calculating the 2.5th and 97.5th percentiles of the generated statistics.
Example of Bootstrap Method for Standard Deviation
- For Murphy's weight, using the weights collected, the bootstrap method results in a standard deviation confidence interval of ( (0.190, 0.399) ).
- Interpretation: With 95% confidence, the true standard deviation of Murphy's weight is captured within this interval.
Types of Statistical Inference
- Confidence intervals (CI) provide an estimated range for a population parameter without prior assumptions.
- Significance tests determine if there is sufficient evidence to support a specific claim about a population parameter.
Evidence of Voter Support for a Candidate
- A sample of 200 registered voters indicated that 114 support a candidate, prompting analysis of majority support.
- Assuming the population proportion is ( 0.5 ), the sampling distribution of ( \hat{p} ) is expressed as ( N(p, \sqrt{\frac{p(1-p)}{n}} ) with mean ( p=0.5 ) and standard deviation ( 0.0345 ).
- Observed ( \hat{p}=0.57 ) results in a low probability (2.39%) of occurrence under the assumption ( p=0.5 ), suggesting the true proportion is likely not 0.5.
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Description
These flashcards provide key definitions and concepts for Statistics Exam 2. Terms include sample proportion, sample mean, and population proportion. Familiarize yourself with these concepts to prepare effectively for your exam.
